On plurisubharmonic defining functions for pseudoconvex domains in ${\mathbb{C}}^2$

Abstract

We investigate the question of existence of plurisubharmonic defining functions for smoothly bounded, pseudoconvex domains in ${\mathbb{C}}^2$. In particular, we construct a family of simple counterexamples to the existence of plurisubharmonic smooth local defining functions. Moreover, we give general criteria equivalent to the existence of plurisubharmonic smooth defining functions on or near the boundary of the domain. These equivalent characterizations are then explored for some classes of domains.

arXiv comment32 pgs., included a family of smoothly bounded, pseudoconvex domains which do not admit smooth local defining functions in Section 3; introduced the notion of sesquiconvexity, a new geometric notion sufficient for the existence of plurisubharmonic defining functions in Section 7